Searcharxiv⌕ Search

arXiv subjects

Giusi Vaira

Publications and source records attributed to Giusi Vaira.

At least 19 recordsLinked to original sources
arXiv2026

Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

We construct families of sign-changing solutions for the four-dimensional Brezis--Nirenberg problem \[ -\Delta u=u^3+\varepsilon u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega, \] as $\varepsilon\to0^+$. A Lyapunov--Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green--Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive--negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.

math.AP↗
arXiv2026

Entire solutions to a strongly competitive nonlinear Schr\"odinger system

We build infinitely-many non-radial positive solutions to the Schr\"odinger system \begin{equation*} \left\{\begin{aligned} &-\Delta u_1+u_1=u_1^{{\mathfrak p} }-\Lambda u_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-\Delta u_2+u_2=u_2^{{\mathfrak p} }-\Lambda u_1^{b_1}u_2^{b_2} \ \hbox{in}\ \mathbb R^N\\ \end{aligned}\right. \end{equation*} with sub-critical $\mathfrak p$-growth as $\Lambda \to +\infty$. The profile of each component is the sum of several copies of the positive solution to $-\Delta U+U=U^{{\mathfrak p} }$ in $\mathbb R^N$, centered at suitable {\em peaks} whose mutual distances diverge as $\Lambda$ increases. More precisely, given two concentric regular polygons with $k$ sides and very large radii, the peaks of the first component are arranged along the edges of the {\em outer} polygon, alternated with those of the second component, and along the $k$ rays joining the vertices of the two polygons. To the best of our knowledge, this provides the first example of non-radial positive solutions for strongly competitive Schr\"odinger systems in the whole space.

math.AP↗
arXiv2025

Existence of positive solutions for a class of almost critical problems on an annulus

In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function and of the Robin function on the annulus, we prove that the annulus becomes thinner and thinner when the number of bumps increases for the slightly subcritical case, while the hole of the annulus is very small for the slightly supercritical case.

math.AP↗
arXiv2025

On Brezis-Nirenberg problems: open questions and new results in dimension six

In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left\{\begin{aligned} &-\Delta u = \lambda u+|u|^{2^*-2}u, \quad &\mbox{in}\,\Omega,\\ &u=0,\quad &\mbox{on}\, \partial\Omega, \end{aligned}\right. \end{equation*} where $\Omega $ is a smoothly bounded domain of $\mathbb R^N$ with $N\geq 3$, $\lambda>0$ is a parameter and $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. We first recall the history of the Brezis-Nirenberg problem and then provide new results of it in dimension six. Finally, we also list some open questions on the Brezis-Nirenberg problem.

math.AP↗
arXiv2025

Blow-up phenomena for a boundary Yamabe problem with umbilic boundary

We consider a linear perturbation of the classical geometric problem of prescribing the scalar and the boundary mean curvature problem in a Riemannian manifold with umbilic boundary provided the Weyl tensor is non-zero everywhere. We will deal with the case of negative scalar curvature showing the existence of a positive solutions when $n\geq 8$.

math.AP↗
arXiv2025

Infinitely many non-radial solutions to a critical Choquard equation

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$ where $V(|x|)$ is a bounded, nonnegative and symmetric potential in $\mathbb R^N$ with $N\geq 5$, $0<\mu\leq 4$, $*$ stands for the standard convolution and $2^\star_\mu:=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy - Littlewood - Sobolev inequality. By applying a finite dimensional reduction method we prove that if $r^2V(r)$ has a local maximum point or local minimum point $r_0>0$ with $V(r_0)>0$ then the problem has infinitely many non-radial solutions with arbitrary large energies.

math.AP↗
arXiv2025

Multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

The paper addresses the existence of multi-bubble solutions for the well-known Brezis-Nirenberg problem. Although there is extensive literature on the subject, the existence of solutions that blow up at multiple points in a 4D bounded domain remains an open problem. The goal of the present paper is to resolve this longstanding issue. In particular, we exhibit examples of domains where a large number of multi-bubble solutions exist. Our result can also be seen as the counterpart of the asymptotic analysis carried out by Konig and Laurin in Ann. Inst. H. Poincar\`e C Anal. Non Lin\`eaire, 2024.

math.AP↗
arXiv2025

Construction of bubbling solutions of the Brezis-Nirenberg problem in general bounded domains (I): the dimensions 4 and 5

In this paper, we consider the Brezis-Nirenberg problem $$ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, \Omega,\quad u=0,\quad\mbox{on}\,\, \partial\Omega, $$ where $\lambda\in\mathbb{R}$, $\Omega\subset\mathbb R^N$ is a bounded domain with smooth boundary $\partial\Omega$ and $N\geq3$. We prove that every eigenvalue of the Laplacian operator $-\Delta$ with the Dirichlet boundary is a concentration value of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ by constructing bubbling solutions with precisely asymptotic profiles via the Ljapunov-Schmidt reduction arguments. Our results suggest that the bubbling phenomenon of the Brezis-Nirenberg problem in dimensions $N=4$ and $N=5$ as the parameter $\lambda$ is close to the eigenvalues are governed by crucial functions related to the eigenfunctions, which has not been observed yet in the literature to our best knowledge. Moreover, as the parameter $\lambda$ is close to the eigenvalues, there are arbitrary number of multi-bump bubbing solutions in dimension $N=4$ while, there are only finitely many number of multi-bump bubbing solutions in dimension $N=5$, which are also new findings to our best knowledge.

math.AP↗
arXiv2025

Infinitely many solutions for a boundary Yamabe problem

We consider the classical geometric problem of prescribing the scalar and the boundary mean curvature in the unit ball endowed with the standard Euclidean metric. We will deal with the case of negative scalar curvature showing the existence of infinitely many non-radial positive solutions when the dimension is larger or equal to 5. This is the first result of existence of solutions in the case of negative prescribed scalar curvature problem in higher dimensions.

math.AP↗
arXiv2024

Partially concentrating solutions for systems with Lotka-Volterra type interactions

In this paper we consider the existence of standing waves for a coupled system of $k$ equations with Lotka-Volterra type interaction. We prove the existence of a standing wave solution with all nontrivial components satisfying a prescribed asymptotic profile. In particular, the $k-1$-last components of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. We analyze first in detail the result with three equations since this is the first case in which the coupling has a role contrary to what happens when only two densities appear. We also discuss the existence of solutions of this form for systems with other kind of couplings making a comparison with Lotka-Volterra type systems.

math.AP↗
arXiv2024

Partially concentrating standing waves for weakly coupled Schrödinger systems

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}ψ_{1}=-\frac{\hslash^2}{2m_{1}}Δψ_{1}+ {V_1}(x)ψ_{1}-μ_{1}|ψ_{1}|^{2}ψ_{1}-β|ψ_{2}|^{2}ψ_{1} & \\ i \hslash \partial_{t}ψ_{2}=-\frac{\hslash^2}{2m_{2}}Δψ_{2}+ {V_2}(x)ψ_{2}-μ_{2}|ψ_{2}|^{2}ψ_{2}-β|ψ_{1}|^{2}ψ_{2},& \end{cases} \] where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_{1}\sim \hslash^2\to0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.

math.AP↗
arXiv2023

Nodal cluster solutions for the Brezis-Nirenberg problem in dimensions $N\geq 7$

We show that the classical Brezis-Nirenberg problem $$Δu + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad Ω, \quad u= 0 , \quad {\mbox {on}} \quad \partial Ω$$ admits nodal solutions clustering around a point on the boundary of $Ω$ as $\varepsilon \to 0$, for smooth bounded domains $Ω\subset \mathbb{R}^N $ in dimensions $N\geq 7$.

math.AP↗
arXiv2023

Positive Blow-up Solutions for a Linearly Perturbed Boundary Yamabe Problem

We consider the problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$ dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature $K$ and boundary mean curvature $H$ of arbitrary sign which are non-constant and $\mathfrak D_n=\sqrt{n(n-1)}{|K|}^{-1/2}>1$ at some point of the boundary. It is known that this problem admits a positive mountain pass solution if $n=3$, while no existence results are known for $n\geq 4$. We will consider a perturbation of the geometric problem and show the existence of a positive solution which blows-up at a boundary point which is critical for both prescribed curvatures.

math.AP↗
arXiv2022

Phase separating solutions for two component systems in general planar domains

In this paper we consider a two component system of coupled non linear Schrödinger equations modeling the phase separation in the binary mixture of Bose-Einstein condensates and other related problems. Assuming the existence of solutions in the limit of large interspecies scattering length $β$ the system reduces to a couple of scalar problems on subdomains of pure phases. Here we show that given a solution to the limiting problem under some additional non degeneracy assumptions there exists a family of solutions parametrized by the parameter $β\gg 1$.

math.AP↗
arXiv2022

Clustering phenomena in low dimensions for a boundary Yamabe problem

We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if $n=3$ all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions $4\leq n\leq 7$. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a local minimizer of the norm of the trace-free second fundamental form of the boundary.

math.AP↗
arXiv2020

A complete scenario on nodal radial solutions to the Brezis Nirenberg problem in low dimensions

In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -Δu=|u|^{2^*-2}u+λu&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball of $\R^N$. We compute the asymptotics of the solution $u$ as well as $||u||_\infty$, its first zero and other relevant quantities as $ł$ goes to a critical value $\barł$. Also the sign of $ł-\barł$ is established in all cases. This completes an analogous analysis for $N\ge7$ given in [EGPV].

math.AP↗